SolvedGraph theory
Four colour theorem
Every map in the plane can be coloured with four colours so that neighbouring regions differ.
Exact statement
Every planar graph has chromatic number at most four.
Status
Solved by Kenneth Appel and Wolfgang Haken (1976) with a computer-assisted proof; formally verified by Georges Gonthier (2005).
Posed: 1852, Francis Guthrie.
In plain words
Regions that share a border must get different colours; regions meeting only at a point may share one. Five colours were easy to prove enough (Heawood, 1890); four needed checking almost 2,000 configurations by computer, the first major theorem proved that way.
How it was solved
- 1890: Heawood proves the five colour theorem.
- 1976: Appel and Haken announce a computer-assisted proof.
- 1997: Robertson, Sanders, Seymour and Thomas give a simpler computer proof.
- 2005: Gonthier verifies the proof in the Coq proof assistant.
What our solver can and cannot do
The Four colour theorem is settled, so the solver shows the result and its history. It can check concrete examples, but the full proof lives in the references below. Open the solver.
Frequently asked questions
- What does the Four colour theorem say?
- Every planar graph has chromatic number at most four.
- Has the Four colour theorem been solved?
- Solved by Kenneth Appel and Wolfgang Haken (1976) with a computer-assisted proof; formally verified by Georges Gonthier (2005).
- Can MathGPT solve the Four colour theorem?
- MathGPT shows the statement, who settled it and how, with references; the proof itself is in the cited papers.
References
- K. Appel and W. Haken, Illinois J. Math. 21 (1977).
- G. Gonthier, 'Formal proof — the four-color theorem', Notices of the AMS 55 (2008).